The well-known result of S. Mazurkiewicz that the simple closed curve is the only nondegenerate locally connected plane homogeneous continuum is extended to generalized homogeneity with respect to some other classes of mappings. Several open problems in the area are posed.
Interrelations between three concepts of terminal continua and their behaviour, when the underlying continuum is confluently mapped, are studied.
Necessary and sufficient conditions are found in the paper for a mapping between continua to be monotone, confluent, semi-confluent, joining, weakly confluent and pseudo-confluent. Three lists of these conditions are presented. Two are formulated in terms of components and of quasi-components, respectively, of connected closed subsets of the range space, while the third one in terms of connectedness between subsets of the domain space. Some basic relations concerning these concepts are studied.
A concept of an absolute end point introduced and studied by Ira Rosenholtz for arc-like continua is extended in the paper to be applied arbitrary irreducible continua. Some interrelations are studied between end points, absolute end points and points at which a given irreducible continuum is smooth.
Continua that are approximative absolute neighborhood retracts (AANR’s) are characterized as absolute terminal retracts, i.e., retracts of continua in which they are embedded as terminal subcontinua. This implies that any AANR continuum has a dense arc component, and that any ANR continuum is an absolute terminal retract. It is proved that each absolute retract for any of the classes of: tree-like continua, -dendroids, dendroids, arc-like continua and arc-like -dendroids is an approximative absolute...
It is shown that for every numbers there is a strongly self-homeomorphic dendrite which is not pointwise self-homeomorphic. The set of all points at which the dendrite is pointwise self-homeomorphic is characterized. A general method of constructing a large family of dendrites with the same property is presented.
Conditions are investigated that imply noncontractibility of curves. In particular, a plane noncontractible dendroid is constructed which contains no homotopically fixed subset. A new concept of a homotopically steady subset of a space is introduced and its connections with other related concepts are studied.
Sufficient as well as necessary conditions are studied for a dendrite or a dendroid to be homogeneous with respect to monotone mappings. The obtained results extend ones due to H. Kato and the first named author. A number of open problems are asked.
Interrelations between smoothness of a continuum at a point, pointwise smoothness, the property of Kelley at a point and local connectedness are studied in the paper.
It is shown that for every two countable ordinals and with there exist -dendroids and whose depths are and respectively, and a monotone retraction from onto . Moreover, the continua and can be either both arclike or both fans.
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